On the number of total prime factors of an odd perfect number

Abstract

Let $N$ be an odd perfect number. Let $\omega (N)$ be the number of distinct prime factors of $N$ and let $\Omega (N)$ be the total number of prime factors of $N$. We prove that if $(3,N)=1$, then $\frac{302}{113}\omega -\frac{286}{113}\le \Omega .$ If $3|N$, then $\frac{66}{25}\omega -5\le \Omega .$ This is an improvement on similar prior results by the author which was an improvement of a result of Ochem and Rao. We also establish new lower bounds on $\omega (N)$ in terms of the smallest prime factor of $N$ and establish new lower bounds on $N$ in terms of its smallest prime factor.

55 pages. This version has the proofs for the lemmas at the end of section 1 included, along with some auxiliary results necessary for those proofs